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CTP-TAMU-06/93

Twenty Years of the Weyl Anomaly †

M. J. Duff ‡

Center for Theoretical PhysicsPhysics Department

Texas A & M UniversityCollege Station, Texas 77843

ABSTRACT

In 1973 two Salam proteges (Derek Capper and the author) discoveredthat the conformal invariance under Weyl rescalings of the metric tensorgµν(x) → Ω2(x)gµν(x) displayed by classical massless field systems in interac-tion with gravity no longer survives in the quantum theory. Since then theseWeyl anomalies have found a variety of applications in black hole physics,cosmology, string theory and statistical mechanics. We give a nostalgic re-view.

CTP/TAMU-06/93July 1993

†Talk given at the Salamfest, ICTP, Trieste, March 1993.‡ Research supported in part by NSF Grant PHY-9106593.

When all else fails, you can always tell the truth.

Abdus Salam

1 Trieste and Oxford

Twenty years ago, Derek Capper and I had embarked on our very first post-

docs here in Trieste. We were two Salam students fresh from Imperial College

filled with ideas about quantizing the gravitational field: a subject which at

the time was pursued only by mad dogs and Englishmen. (My thesis title:

Problems in the Classical and Quantum Theories of Gravitation was greeted

with hoots of derision when I announced it at the Cargese Summer School

en route to Trieste. The work originated with a bet between Abdus Salam

and Hermann Bondi about whether you could generate the Schwarzschild

solution using Feynman diagrams. You can (and I did) but I never found

out if Bondi ever paid up.)

Inspired by Salam, Capper and I decided to use the recently discovered

dimensional regularization1 to calculate corrections to the graviton propaga-

tor from closed loops of massless particles: vectors [1] and spinors [2], the

former in collaboration with Leopold Halpern. This involved the self-energy

1Dimensional regularization had just been invented by another Salam student andcontempory of ours, Jonathan Ashmore [3], and independently by Bollini and Giambiagi [4]and by ’t Hooft and Veltman [5]. I briefly shared a London house with Jonathan Ashmoreand Fritjof Capra. Both were later to leave physics, as indeed was Derek Capper. Ashmorewent into biology, Capper into computer science and Capra into eastern mysticism (adecision in which, as far as I am aware, Abdus Salam played no part).

1

insertion

Πµνρσ(p) =∫dnxeipx < Tµν(x)Tρσ(0) >|gµν=δµν (1)

where n is the spacetime dimension and Tµν(x) the energy-momentum tensor

of the massless particles. One of our goals was to verify that dimensional

regularization correctly preserved the Ward identity

pµΠµνρσ(p) = 0 (2)

that follows as a consequence of general covariance. If we denote by ǫ the

deviation from the physical spacetime dimension one is interested in, and

expand about ǫ = 0, we obtain

Πµνρσ =1

ǫΠµνρσ(pole) + Πµνρσ(finite) (3)

Capper and I were able to verify that the pole term correctly obeyed the

Ward identity

pµΠµνρσ(pole) = 0 (4)

and that the infinity could then be removed by a generally covariant coun-

terterm. We checked that the finite term also obeyed the identity

pµΠµνρσ(finite) = 0 (5)

and hence that there were no diffeomorphism anomalies.2

2Had we looked at closed loops of Weyl fermions or self-dual antisymmetric tensors in2 mod 4 dimensions, and had we been clever enough, we would have noticed that thisWard identity breaks down. But we didn’t and we weren’t, so this had to wait anotherten years for the paper by Alvarez-Gaume and Witten [6].

2

We were also aware that since the massless particle systems in question

were invariant under the Weyl transformations 3 of the metric

gµν(x) → Ω2(x)gµν(x) (6)

together with appropriate rescalings of the matter fields, this implied that

the stress tensors in (1) should be traceless and hence that the self energy

should also obey the trace identity

Πµµρσ(p) = 0 (7)

We verified that the pole term was OK:

Πµµρσ(pole) = 0 (8)

consistent with our observation that the counterterms were not only generally

covariant but Weyl invariant as well. For some reason, however, I did not get

around to checking the finite term until Christmas of 73 by which time I was

back in England on my second postdoc, in Oxford, where Dennis Sciama was

gathering together a group of quantum gravity enthusiasts. To my surprise,

I found that

Πµµρσ(finite) 6= 0 (9)

I contacted Derek and he confirmed that we hadn’t goofed. The Weyl in-

variance (6) displayed by classical massless field systems in interaction with

gravity, first proposed by Hermann Weyl in 1918 [7, 8, 9], no longer survives

3For fermions this is true ∀n; for vectors only for n = 4

3

in the quantum theory! We rushed off a paper [10] to Nuovo Cimento (How

times have changed!).

I was also able to announce the result at The First Oxford Quantum

Gravity Conference, organised by Isham, Penrose and Sciama, and held at

the Rutherford Laboratory in February 74 [11]. Unfortunately, the announce-

ment was somewhat overshadowed because Stephen Hawking chose the same

conference to reveal to an unsuspecting world his result [12] that black holes

evaporate! Ironically, Christensen and Fulling [13] were subsequently to show

that in two spacetime dimensions the Hawking effect is due entirely to the

trace anomaly. Two dimensional black holes, and the effects of the Weyl

anomaly in particular [14, 15, 16], are currently enjoying a revival of interest.

2 Anomaly? What anomaly?

Some cynic once said that in order for physicists to accept a new idea, they

must first pass through the following three stages:

(1) It’s wrong

(2) It’s trivial

(3) I thought of it first.

In the case of the Weyl anomaly, however, our experience was that (1)

and (2) got interchanged. Being in Oxford, one of the first people we tried to

impress with our new anomaly was J. C. Taylor who merely remarked “Isn’t

that rather obvious?”. In a sense, of course, he was absolutely right. He

4

presumably had in mind the well-known result that theories which are Weyl

invariant in n-dimensional curved space are automatically invariant under the

conformal group SO(2, n) in the flat space limit, which implies in particular

that the dilatation current Dµ ≡ xνT µν is conserved:

∂µDµ = T µ

µ + xν∂µTµ

ν = T µµ = 0 (10)

Moreover, one already knew from the work of Coleman and Jackiw [17] that

such flat-space symmetries suffered from anomalies. Of course, the Weyl

invariance (6), in contrast to the conformal group, is a local symmetry for

which there is therefore no Noether current. Nevertheless, perhaps one should

not be too surprised to discover that there is a curved space generalization in

the sense of a non-vanishing trace for the stress tensor. Consequently, Capper

and I were totally unprepared for the actual response of the rest of the physics

community: NO-ONE BELIEVED US! To be fair, we may have put some

people off the scent by making the correct, but largely irrelevant, remark

that at one loop the anomalies in the two-point function could be removed

for n 6= 2 by adding finite local counterterms [10]. So we wrote another paper

[18] stressing that the anomalies were real and could not be ignored, but to

no avail. To rub salt in the wounds, among those dismissing our result as

spurious were physicists for whom we youngsters had the greatest respect.

5

First the Americans:

...the finite Wreg that is left behind by Schwinger’s method, after the infini-

ties have been split off, is both coordinate invariant and conformally invariant,

DeWitt [19]

Something is wrong, Christensen [20]

The form of the conformal anomaly in the trace of the stress tensor pro-

posed by a number of authors violates axiom 5, Wald [21]

Thus we find no evidence of the conformal trace anomalies reported by a

number of other authors, Adler, Lieberman and Ng [22],

then the Europeans:

Conformal anomalies in a conformally invariant theory do not arise...,

Englert, Gastmans and Truffin [23]

There are important, physically relevant differences: most noticeable, nor-

malized energy-momentum tensors do not possess a ‘trace anomaly’, Brown

and Ottewill [24],

especially the Russians:

The main assumption of our work is that a regularization scheme exists

which preserves all the formal symmetry properties (including the Weyl sym-

metry)...Therefore we hope dimensional regularization will give no anoma-

lies.., Kallosh [25]

It turns out that conformal anomalies, discovered in gravitating systems,

are not true anomalies, since in modified regularizations they do not arise....

6

The above point of view on conformal anomalies is shared by Englert et al.,

Fradkin and Vilkovisky [26]

The presence or absence of the conformal anomaly depends on the choice

made between two existing classes of covariant regularizations, Vilkovisky

[27]

and, to be democratic, let us not forget the Greeks:

The above method of regularization and renormalization preserves the

Ward identities...and trace anomalies do not arise, Antoniadis and Tsamis

[28]

There exists a regularization scheme which preserves both general coordi-

nate invariance and local conformal invariance (Englert et al)..., Antoniadis,

Iliopoulos and Tomaras [29]

In fact one needs a regularization scheme which preserves the general

coordinate and Weyl invariance...such a scheme exists (Englert 4 et al), An-

toniadis, Kounnas and Nanopoulos [30].

4If one starts with a classically non-Weyl invariant theory (e.g. pure Einstein grav-ity) and artificially makes it Weyl invariant by introducing via a change of variablesg′

µν(x) = e2σ(x)gµν(x) an unphysical scalar spurion σ(x), then unitarity guarantees that

no anomalies can arise because this artificial Weyl invariance of the quantum theory,g′

µν(x) → Ω2(x)g′

µν(x) with e2σ(x) → Ω2(x)e2σ(x), is needed to undo the field redefinition

and remove the spurious degree of freedom. Professor Englert informed me in Triestethat this is what the authors of [23] had in mind when they said that anomalies do notarise. Let us all agree therefore that many of the apparent contradictions are due to thismisunderstanding.

7

3 London

As chance would have it, my third postdoc brought me to King’s College,

London, at the same time as Steve Christensen, Paul Davies, Stanley Deser,

Chris Isham and Steve Fulling. Bill Unruh was also a visitor. It was des-

tined therefore to become be a hot-bed of controversy and activity in Weyl

anomalies. Provoked by Christensen and Fulling, who had not yet been con-

verted, Deser, Isham and I decided to write down the most general form of

the trace of the energy-momentum tensor in various dimensions [31]. By

general covariance and dimensional analysis, it must take the following form:

For n=2,

gαβ < Tαβ >= aR, (11)

where a is a constant. For n=4,

gαβ < Tαβ >= αR2 + βRµνRµν + γRµνρσR

µνρσ + δ R + cFµνaF µνa, (12)

where α, β, γ, δ and c are constants. (In (12) we have allowed for the pos-

sibility of an an external gauge field in addition to the gravitational field.)

For n = 6, gαβ < Tαβ > would have to be cubic in curvature and so on. (At

one-loop, and ignoring boundary terms, there is no anomaly for n odd). I

showed these expressions to Steve Christensen, with whom I was sharing an

office, and asked him if he had seen anything like this before. He immediately

became very excited and told me that these were precisely the Schwinger-

Dewitt bn coefficients. These are the t-independent terms that appear in

8

the asymptotic expansion of the heat kernel of the appropriate differential

operators ∆:

Tr e−∆t ∼∞∑

k=0

Bkt(k−n)/2 t→ 0+ (13)

where

Bk =∫dnx bk (14)

For example, if ∆ is the conformal Laplacian

∆ = − +R

6(15)

then b4 is given by

b4 =1

180(4π)2[−RµνR

µν +RµνρσRµνρσ + R] (16)

This was the road to Damascus for Steve as far as Weyl anomalies were

concerned and, like many a recent convert, he went on to become their most

ardent advocate 5. This was also the beginning of a very fruitful collaboration

between the two of us. The significance of my paper with Deser and Isham

was that, with the exception of the R term in (12), none of the above anoma-

lies could be removed by the addition of finite local counterterms (hence the

title Non-local Conformal Anomalies) and thus this laid to rest any lingering

5 A delightful set of reminiscences on the Weyl anomaly by Steve Christensen can alsobe found in Bryce DeWitt’s festschrift [32]. During their stay at King’s, he and Fullingshared a flat in the London borough of Ealing (home of the famous Ealing Comedy movies).According to Christensen, the connection between trace anomalies and the Hawking effectoccurred, Archimedes-like, to Fulling while taking a bath. He did not run through thestreets of Ealing shouting “Eureka”, but did run upstairs to the payphone to tell BillUnruh.

9

doubts about the inevitability of Weyl anomalies (or, at least, it should have

done). By this time, or shortly afterwards, the Hawking radiation experts at

King’s and elsewhere were arriving at the same conclusion[13, 33, 34, 35, 36]

as in fact was Hawking himself [37]. There followed a series of papers calcu-

lating the numerical coefficients in (11) and (12) and confirming that these

were indeed just the bn coefficients [38, 39, 40, 41, 42]. These results were

further generalized to self-interacting theories [43, 44, 45, 46, 47].

One day about this time I answered the phone in my office only to hear

those five words most designed to instill fear and trembling into the heart

of a young postdoc: “Hi. This is Steven Weinberg”. Pondering on the

non-renormalizability problem, Weinberg had become interested in quan-

tum gravity in 2 + ǫ dimensions [48, 49]. Inspired by workers in statistical

mechanics, who frequently work with non-renormalizable field theories but

who nevertheless manage to extract sensible predictions, Weinberg wondered

whether this might be true for gravity: was the theory “asymptotically safe”?

The answer seemed to rely on the sign of the two-dimensional trace anomaly

i.e. on the constant a in (11). Accordingly, Weinberg set n=2 in the n-

dimensional calculations of [2] and concluded that fermions had the wrong

sign:

a =1

24π(17)

He repeated our calculation for scalars himself and found the same sign and

magnitude (consistent with the observation that in two-dimensions there is

10

a bose-fermi equivalence, and consistent with the black hole calculations

[33, 13]). In the case of vector bosons, however, he found from [1] that there

was a sign flip. His question was simple but crucial: did I agree with him

or could there be an overall sign error? Not wanting to be the victim of

Weinberg’s wrath should I get it wrong, I spent several frantic days and

sleepless nights checking and rechecking the calculations. Those who have

ever chased a minus sign and those who know Steve Weinberg will appreciate

my discomfort! In fact I agreed. Unfortunately, asymptotic safety became

asymptotically unpopular but my contact with Weinberg later led to a very

fruitful semester in Austin, and to my continuing affection for the state of

Texas.

The scalar terms of order n/2 in the curvature which appear in the n-

dimensional gravitational trace anomaly are reminiscent of the pseudoscalar

terms of order n/2 in the curvature which appear in the n-dimensional grav-

itational axial anomaly, as calculated by Delbourgo and Salam [50]. I was

musing on this shortly after moving across town to Queen Mary College,

when I saw a paper by Eguchi and Freund [51] on the then new and excit-

ing topic of gravitational instantons. They considered the two topological

invariants, the Pontryagin number, P , and the Euler number, χ, and posed

the question: To what anomalies do P and χ contribute? In the case of the

Pontryagin number, they were able to answer this question by relating P

to the integrated axial anomaly; in the case of the Euler number, however,

11

they found no anomaly. I therefore wrote a short note [52] relating χ to the

integrated trace anomaly. As described section 6, this result was later to

prove important in the two-dimensional context of string theory, where

χ =1

4π

∫d2x

√gR (18)

and hence from (11)

1

4π

∫d2x

√ggαβ < Tαβ >= aχ (19)

Unfortunately, the referee’s vision did not extend that far and the paper was

rejected. Rather than resubmit it, I decided to incorporate the results into

a larger paper [53] which re-examined the Weyl anomaly in the light of its

applications to the Hawking effect, to gravitational instantons, to asymptotic

freedom and Weinberg’s asymptotic safety. In the process, I discovered that

the constants α, β, γ and δ are not all independent but obey the constraints

4α + β = α− γ = −δ (20)

In other words, the gravitational contribution to the anomaly depends on

only two constants (call them b and b′) so that (12) may be written

gαβ < Tαβ >= b(F +2

3R) + b′G+ cH (21)

where

F = RµνρσRµνρσ − 2RµνR

µν +1

3R2, (22)

G = RµνρσRµνρσ − 4RµνR

µν +R2 (23)

12

and

H = FµνaF µνa (24)

In four (but only four) dimensions

F = CµνρσCµνρσ (25)

where Cµνρσ is the Weyl tensor, and G is proportional to the Euler number

density

G =∗ Rµνρσ∗Rµνρσ (26)

where ∗ denotes the dual. Note the absence of an R2 term in (21). This

result was later rederived using the Wess-Zumino consistency conditions [54,

55, 56, 57, 58]. Furthermore, the constants a, b, b′ and c are those which

determine the counterterms

∆L =a

ǫ

√gR n = 2 (27)

∆L =1

ǫ

√g(bF + b′G+ cH) n = 4 (28)

(and hence the renormalization group β functions) at the one-loop level.

The Euler number counterterms are frequently ignored on the grounds that

they are total divergences, but will nevertheless contribute in spacetimes of

non-trivial topology. We emphasize that the above results are valid only for

theories which are classically conformally invariant (e.g. Maxwell/Yang-Mills

for n=4 only, and conformal scalars and massless fermions for both n=2 and

n=4). For other theories (e.g. Maxwell/Yang-Mills for n=2, pure quantum

13

gravity for n=4, or any theory with mass terms) the “anomalies” will still

survive, but will be accompanied by contributions to gαβ < Tαβ > expected

anyway through the lack of conformal invariance. Since the anomaly arises

because the operations of regularizing and taking the trace do not commute,

the anomaly in a theory which is not classically Weyl invariant may be defined

as:

Anomaly = gαβ < Tαβ >reg − < gαβTαβ >reg (29)

Of course, the second term happens to vanish when the classical invariance

is present.

Note that in an expansion about flat space with gµν = δµν + hµν , R is

O(h), so it is sufficient to calculate the two-point function as in [10] to fix

the a coefficient of R in (11). For n=4, R is O(h) while F and G are

O(h2). Nevertheless, because of the constraint (20), a calculation of the two-

point function is again sufficient to fix the b coefficient of CµνρσCµνρσ in (21),

notwithstanding the ability to remove the R piece by a finite local countert-

erm, and notwithstanding some contrary claims in the literature. Indeed, this

is how the coefficients of the Weyl invariant CµνρσCµνρσ counterterms were

first calculated [1, 10].

Explicit calculations [1, 10, 2, 59, 13, 38, 36, 41, 39, 53] yield

b =1

120(4π)2[NS + 6NF + 12NV ] (30)

b′ = − 1

360(4π)2[NS + 11NF + 62NV ] (31)

14

where NS, NF and NV are respectively the numbers of scalars, spin 1/2

Dirac fermions and vectors in the theory. Note that the contributions to

the b coefficient are all positive, as they must be by spectral representation

positivity arguments [2, 59, 58] on the vacuum polarization proper self-energy

part in four dimensions.

I also noted that the non-local effective action responsible for the anoma-

lies would contain a term

Seff ∼∫dnx

√gR (n−4)/2R (32)

By setting n = 2 one obtains what what later to be known as the Polyakov

action, discussed in section 6.

4 Cosmology

The role of the Weyl anomaly in cosmology seems to fall into the following

categories: inflation in the early universe, the vanishing of the cosmological

constant in the present era, particle production and wormholes.

The first is reviewed in papers by Grischuk and Zeldovitch [60] and Olive

[61]. Consider the semi-classical Einstein equations

Rµν −1

2gµνR = 8πG < Tµν > (33)

where < Tµν > is the effective stress-tensor induced by quantum loops. In

the inflationary phase, the geometry will be that of De Sitter space. But the

15

trace anomaly for De Sitter completely determines the energy momentum

because it must be a multiple of the metric by the symmetry 6:

< Tµν >=1

4gµνg

αβ < Tαβ > (34)

The idea that the trace anomaly might also have a bearing on the van-

ishing of the cosmological constant is a recurring theme [62, 28, 63, 64, 65,

66, 67]. According to Tomboulis [64], models where the cosmological con-

stant relaxes dynamically to zero via some dilaton-like scalar field suffer from

an unnatural fine-tuning of the parameters. This problem can be cured, he

claims, if the Wess-Zumino functional induced by the conformal anomaly

is included. A similar approach has been taken by Antoniadis, Mazur and

Mottola [65, 66, 67], who argue that four-dimensional gravity is drastically

modifed at distances larger than the horizon scale, due to the large infrared

quantum fluctuations of the conformal part of the metric, whose dynamics

is governed by an effective action induced by the trace anomaly, analogous

to the Polyakov action in two dimensions. Apparently, this leads to a con-

formally invariant phase in which the effective cosmological term necessarily

vanishes. See also [68, 62, 69, 70] for a discussion of the cosmological con-

stant/trace anomaly connection in the context of spacetime foam [68].

With regard to particle production, Parker [71] has used the trace anomaly

to argue that there is no particle production by a gravitational field if space-

6The trace anomaly also determines the energy momentum completely for a two di-mensional black hole and in the four dimensional case it determines it up to one functionof position [13].

16

time is conformally flat and quantum fields are conformally coupled, but this

has recently been challenged by Massacand and Schmid [72].

Finally, Grinstein and Hill [73] and also Ellis, Floratos and Nanopoulos

[74] have claimed that in Coleman’s wormhole scenario [75], it is the trace

anomaly that controls the behavior of fundamental coupling constants, par-

ticle masses, mixing angles, etc.

5 Supersymmetry

The Weyl anomaly acquires a new significance when placed in the context of

supersymmetry. In particular, Ferrara and Zumino [76] showed that the trace

of the stress tensor T µµ, the divergence of the axial current ∂µJ

µ5, and the

gamma trace of the spinor current γµSµ form a scalar supermultiplet. There

followed a good deal of activity in calculating the corresponding anomalies

in global supersymmetry.

In the period 1977-9, Christensen and I found ourselves in Boston: he

at Harvard; I at Brandeis. We decided to look at these anomalies in super-

gravity. Since the supermultiplets involve fields eµa, ψµ, Aµ, χ, φ with spins

2, 3/2, 1, 1/2, 0 we first determined the axial and trace anomalies for fields

of arbitary spin [77, 78]. See also [79, 80, 81, 82, 83]. One of our main mo-

tivations was to calculate the gravitational spin 3/2 axial anomaly 7 which

7Our result, that the Rarita-Schwinger anomaly was −21 times the Dirac, generatedalmost as much incredulity as did the Capper-Duff Weyl anomaly, but that’s anotherfestschrift.

17

Field 360A N = 8 N = 4 N = 4supergravity supergravity Y ang −Mills

eaµ 848 1 1 0ψµ −233 8 4 0Aµ −52 28 6 1χ 7 56 4 4φ 4 63 1 6φµν 364 7 1 0φµνρ −720 1 0 0

A = 0 A = 0 A = 0

Table 1: Vanishing anomaly in N=8 and N=4 supermultiplets.

was at the time unknown. Shortly afterwards (by which time I had come full

circle and was back at Imperial College) van Nieuwenhuizen and I noted [84]

that the gravitational trace anomaly for a field of given spin could depend on

the field representation. Thus a rank two gauge φµν field yielded a different

result from a scalar φ, even though they are dual to one another. Similarly,

a rank three gauge field φµνρ yielded a non-zero result, even though it is dual

to nothing. However, these differences showed up only in the coefficient of

the topological Euler number term.

At one loop in supergravity after going on-shell, we may write the anomaly

as

gαβ < Tαβ >=A

32π2∗Rµνρσ

∗Rµνρσ (35)

so that when (21) applies, A = 32π2(b+ b′). The contributions to the A co-

efficient from the various fields are shown in Table 1. Note that the fermions

18

are Majorana. The significance of these results lies in their application to the

D = 4, N = 8 supergravity obtained by dimensional reduction from Type II

supergravity inD = 10 and theD = 4, N = 4 supergravity-Yang-Mills super-

multiplets which arise from dimensional reduction from N=1 supergravity-

Yang-Mills in D=10. As we can see from field content given in the last three

columns of Table 1, the combined anomaly exactly cancels 8. I have singled

out these supermultiplets because these are precisely the field theory limits

of the toroidally compactified Type II and heterotic superstrings. Indeed,

these results have recently been confirmed in a direct string calculation by

Antoniadis, Gava and Narain [113].

Another application of the trace anomaly in the context of supersymmetry

concerned the gaugedN -extended supergravities which exhibit a cosmological

constant proportional to the gauge coupling e. By calculating the Weyl

anomaly in the presence of a cosmological constant [62, 70], therefore, one

can determine the renormalization group beta function β(e). One finds,

remarkably, that the one-loop β function vanishes for N > 4 [69].

See [87] for a review of Weyl anomalies in supergravity, and [88] for those

in conformal supergravity.

8Curiously enough, before the gravitino contribution to the anomaly was calculatedexplicitly, D’Adda and Di Vecchia [85] attempted to deduce it by assuming that the totalanomaly cancels in N = 4 supergravity. This was a good idea but they reached the wrongconclusion by working with the dual formulation with two φ fields instead of the stringyversion with one φ and one φµν obtained by dimensional reduction.

19

6 The string era

The history of the Weyl anomaly took a new turn with the advent of string

theory. The emphasis shifted away from four dimensional spacetime to the

two dimensions of the string worldsheet. In particular, in two very influential

1981 papers, Polyakov [89, 90] showed that the critical dimensions of the

string correspond to the absence of the two dimensional Weyl anomaly. In

the first quantized theory of the bosonic string, one starts with a Euclidean

functional integral over the worldsheet metric γij[ξ], i, j = 1, 2, and spacetime

coordinatesXµ[ξ], µ = 0, 1, ..., D−1, where ξi are the worldsheet coordinates.

Thus

e−Γ =∫

Dγ DX

V ol(Diff)e−S[γ,X] (36)

where

S[γ,X] =1

4πα′

∫d2ξ

√γγij∂iX

µ∂jXνηµν (37)

As showed by Polyakov, the Weyl anomaly in the worldsheet stress tensor is

given by

γij < Tij >=1

24π(D − 26)R(γ) (38)

The contribution of the D scalars follows from (17) while the −26 arises

from the diffeomorphism ghosts that must be introduced into the functional

integral. In the case of the fermionic string, the result is

γij < Tij >=1

16π(D − 10)R(γ) (39)

20

Thus the critical dimensions D = 26 and D = 10 correspond to the preser-

vation of the two dimensional Weyl invariance γij → Ω2(ξ)γij. One may

wonder how Polyakov addressed the controversy of the previous eight years

described in section 3. Well he didn’t, but merely remarked “This is the

well-known trace anomaly”.

Previously, the critical dimensions had been understood from the central

charge c of the Virasoro algebra [91]

[Ln, Lm] = (n−m)Ln+m +c

12n(n2 − 1)δn,−m (40)

where the Ln are the coefficents in a Laurent expansion of the stress tensor,

namely

T (z) =∑n∈Z

Lnz−n−2 (41)

where T = Tzz and z ≡ exp(ξ0 + iξ1). Thus this established a connection

between the two dimensional Weyl anomaly and the central charge of the

Virasoro algebra (to be precise, c = 24πa in equation (11)) ; a result which

spawned the whole industry of conformal field theory in the context of strings.

See, for example, Alvarez-Gaume [92]. In fact, when writing (41), one usually

assumes that Tij is traceless, which forces the anomaly to show up as a

diffeomorphism anomaly, but the results are entirely equivalent [93].

Polyakov went on to describe what happens in non-critical string theory

when the Weyl invariance is lost, and the metric conformal mode propagates.

21

In this case, the two dimensional effective action is given by

Seff ∼∫d2ξ

√γ[R −1R + µ] (42)

where we have allowed for a worldsheet cosmological term produced by quan-

tum corrections. If we now separate out the conformal mode σ and let

γij → eσγij, we obtain the Liouville action

SL[σ] =∫d2ξ

√γ(

1

2γij∂iσ∂jσ +Rσ + µeσ) (43)

This is the starting point for much of non-critical string theory.

The role of the Weyl anomaly becomes even more interesting when we

allow for the presence of the spacetime background fields, as shown by Callan

et al [94] and Fradkin and Tseytlin [95]. In the case of the bosonic string,

for example, the worldsheet action takes the form

S =1

2πα′

∫d2ξ

1

2[√γγij∂iX

µ∂jXνGµν(X) + ǫij∂iX

µ∂jXνBµν(X)]

+1

4π

∫d2ξ

√γR(γ)Φ(X). (44)

corresponding to background fields Gµν(X), Bµν(X) and Φ(X). Now the

anomaly may be written as

1

2πα′γij < Tij >= βΦ√γR(γ) + βG

µν√γγij∂iX

µ∂jXν + βB

µνǫij∂iX

µ∂jXν

(45)

The absence of the Weyl anomaly thus means the vanishing of the β functions,

which to lowest order turn out to be [94]

0 = βGµν = Rµν −

1

4Hµ

λσHνλσ + 2∇µ∇νΦ +O(α′) (46)

22

0 = βBµν = ∇λHµν

λ − 2∇λΦHλ

µν +O(α′) (47)

0 = 16π2βΦ =D − 26

3α′+ [4(∇Φ)2 − 4∇2Φ − R− 1

12H2] +O(α′2) (48)

But these are nothing but the Einstein-matter field equations that result

from the action

Γeff ∼∫dDx

√−Ge−2Φ[

D − 26

3α′−R − 4(∇Φ)2 − 1

12H2] + ... (49)

The common factor e−2Φ reveals that these terms are tree-level in a string

loop perturbation expansion. If we denote the dilaton vacuum expectation

value by Φ0, then from (18) the classical action (44) yields a term

e−χΦ0 = e−2(1−L)Φ0 (50)

in the functional integral, where L counts the number of holes in the two-

dimensional Riemann surface, i.e. the number of loops.

That the Einstein equations in spacetime should originate from the van-

ishing of the worldsheet Weyl anomaly is perhaps the most remarkable result

in our story.

7 Current Problems

New techniques for calculating Weyl anomalies continue to appear. Fujikawa

[96, 97] has pioneered the functional integral approach where the origin of

the lack of quantum Weyl invariance in a classically invariant theory may be

attributed to a non-invariant measure in the functional integral. Ceresole,

23

Pizzochero, and van Nieuwenhuizen [98] have reproduced these results using

flat-space plane waves. Bastianelli and van Nieuwenhuizen [99, 100] have

applied to Weyl anomalies the quantum mechanical approach, first used by

Alvarez Gaume and Witten [6] in the context of axial anomalies.

Much of the current interest in the Weyl anomaly resides not only in

high-energy physics and general relativity but in statistical mechanics. See,

for example [92, 101, 102, 103, 104, 105, 106]. A particularly powerful result

is Zamolodchikov’s c-theorem [107], which states that there exists a function

defined on the space of two-dimensional conformal field theories which is

decreasing along renormalization group (RG) trajectories, and is stationary

only at RG fixed points, where its value equals the Virasoro central charge c.

Recently, there has been a good deal of activity by Cardy [108], Osborn [109],

Jack and Osborn [110], Cappelli, Friedan and Latorre [58], Shore [111, 112],

Antoniadis, Mazor and Mottola [86] and Osborn and Petkos [114] attempting

to generalize this theorem to higher dimensions. Whereas from (11) the two

dimensional gravitational anomaly depends on only one number, however,

from (12) the four dimensional one depends on two: the Euler term and the

Weyl term. In higher dimensions, there will always be one Euler term, but

the number of Weyl invariants grows with dimension [115]. Consequently, it

is not clear whether there is a unique way to generalize the theorem nor how

useful such a generalization might be. Cardy and Jack and Osborn focussed

on the Euler number term, whereas Cappelli et al pointed to the positivity

24

of the Weyl tensor term (30) as a more likely guide, but the analysis is still

inconclusive.

On the subject of the Euler term, the results of section 5 are, in fact,

still controversial because the anomaly inequivalence between different field

representations, discussed by van Nieuwenhuizen and myself, was challenged

at the time by Siegel [116] and by Grisaru, Siegel and Zanon [117]. They

found that the traces of the two stress tensors were equivalent . Yet the

recent string results of Antoniadis et al [86] would seem to support our in-

terpretation. Moreover, if it were incorrect, the vanishing of the anomalies

for the N=4 and N=8 multiplets would seem to be a gigantic coincidence.

Nevertheless, to tell the truth (in accordance with Salam’s maxim), I

am still uneasy about the whole thing. The numerical coefficients quoted in

Table 1 were calculated using the b4 coefficients discussed in section 3. The

claim that these correctly describe the trace anomaly is in turn based on an

identity which everyone used to take for granted. See, for example, Hawking

[37]. The identity says that if S[g] is a functional of the metric gµν , then

∫dnx gµν δS[g]

δgµν≡ ∂S[λg]

∂λ|λ=1 (51)

where λ is a constant. If true, it would mean that the integrated trace of the

stress tensor arises exclusively from the non-invariance of the action under

constant rescalings of the metric. However, the Polyakov action provides an

25

obvious counterexample:

Seff ∼∫d2x

√gR −1R (52)

This action is scale invariant, but gives rise to an anomaly proportional to R!

Of course, the integrated anomaly is a purely topological Euler number term,

and it is the topological nature of the action which provides the exception to

the rule. However, since the entire debate over equivalence versus inequiva-

lence devolved precisely on Euler number terms, the use of the identity (51)

in this context makes me feel very uneasy.

The whole question of whether the dual formulations of supergravity yield

the same Weyl anomalies has recently been thrust into the limelight with

the string/fivebrane duality conjecture [118, 119] which states that in their

critical spacetime dimension D = 10, superstrings (extended objects with

one spatial dimension) are dual to superfivebranes (extended objects with

five spatial dimensions [120]). Whereas the two dimensional worldsheet of

the string couples to the rank two field φµν , the six dimensional worldvolume

of the fivebrane couples to the rank six field φµνρλστ . The usual rank two

formulation of D=10 supergravity dimensionally reduces to the N = 4 field

content of Table 1, but the dual rank six formulation reduces to a different

field content with 20 φµνρ and with 14 φ replaced by 14 φµν . If Table 1 is

to be believed, A(φµν) − A(φ) = 1 and A(φµνρ) = −2. Therefore the dual

version has non-vanishing coefficent A = 14 − 40 = −26. This increases my

uneasiness.

26

A possible resolution of this problem may perhaps be found in the recent

paper by Deser and Schwimmer[115] who have re-examined the different ori-

gin of the topological versus Weyl tensor contributions to the anomaly (which

they call Type A and Type B, respectively).

I certainly believe that the final word on this subject has still not been

written.

8 Acknowledgements

I would like to thank my anomalous collaborators Derek Capper, Steve Chris-

tensen, Stanley Deser, Chris Isham, and Peter van Nieuwenhuizen.

Most of all, however, my thanks go to my former supervisor Abdus Salam

who first kindled my interest in quantum gravity and who continues to pro-

vide inspiration to us all as a scientist and as a human being. When the

deeds of great men are recalled, one often hears the cliche “He did not suffer

fools gladly”, but my memories of Salam at Imperial College were quite the

reverse. People from all over the world would arrive and knock on his door to

expound their latest theories, some of them quite bizarre. Yet Salam would

treat them all with the same courtesy and respect. Perhaps it was because

his own ideas always bordered on the outlandish that he was so tolerant of

eccentricity in others; he could recognize pearls of wisdom where the rest of

us saw only irritating grains of sand. As but one example of a crazy Salam

idea, I distinctly remember him remarking on the apparent similarity be-

27

tween the mass and angular momentum relation of a Regge trajectory and

that of an extreme black hole. Nowadays, of course, string theorists will jux-

tapose black holes and Regge slopes without batting an eyelid, but to suggest

this back in the late 1960’s was considered preposterous by minds lesser than

Salam’s.9

Theoretical physicists are, by and large, an honest bunch: occasions when

scientific facts are actually deliberately falsified are almost unheard of. Nev-

ertheless, we are still human and consequently want to present our results

in the best possible light when writing them up for publication. I recall a

young student approaching Abdus Salam for advice on this ethical dilemma:

“Professor Salam, these calculations confirm most of the arguments I have

been making so far. Unfortunately, there are also these other calculations

which do not quite seem to fit the picture. Should I also draw the reader’s

attention to these at the risk of spoiling the effect or should I wait? After all,

they will probably turn out to be irrelevant.” In a response which should be

immortalized in The Oxford Dictionary of Quotations, Salam replied: “When

all else fails, you can always tell the truth”.

Amen.

9Historical footnote: at the time Salam had to change the gravitational constant tomatch the hadronic scale, an idea which spawned his strong gravity; today the fashion isthe reverse and we change the Regge slope to match the Planck scale!

28

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